91Ó°ÊÓ

• The partial fraction expansion of a rational fraction (that is, a fraction with both a polynomial numerator and a polynomial denominator) is an algebraic procedure that involves expressing the fraction as a sum of a polynomial plus one or more fractions with a simpler denominator.
• The partial fraction decomposition is important because it gives methods for a variety of rational function computations, such as explicit antiderivative computations, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms.

The given rational function is $\frac{−\mathrm{s}−7}{(\mathrm{s}+1)(\mathrm{s}−2)}$

Rewrite $\frac{−\mathrm{s}−7}{(\mathrm{s}+1)(\mathrm{s}−2)}$ as a sum of partial fractions as:

$\frac{−\mathrm{s}−7}{(\mathrm{s}+1)(\mathrm{s}−2)}=\frac{\mathrm{A}}{\mathrm{s}+1}+\frac{\mathrm{B}}{\mathrm{s}−2}$

Multiply both sides by the LCD $(\mathrm{s}+1)(\mathrm{s}−2)$as follows:

$−\mathrm{s}−7=\mathrm{A}(\mathrm{s}−2)+\mathrm{B}(\mathrm{s}+1)$

Find the constants as:

For$\mathrm{s}=−1,−(−1)−7=\mathrm{A}(−3)⇒\mathrm{A}=2$.

For $\mathrm{s}=2,−\left(2\right)−7=\mathrm{B}\left(3\right)⇒\mathrm{B}=−3$.

Substitute the value of constants into $\frac{−\mathrm{s}−7}{(\mathrm{s}+1)(\mathrm{s}−2)}=\frac{\mathrm{A}}{\mathrm{s}+1}+\frac{\mathrm{B}}{\mathrm{s}−2}$as follows:

$\begin{array}{c}\frac{−\mathrm{s}−7}{(\mathrm{s}+1)(\mathrm{s}−2)}=\frac{2}{\mathrm{s}+1}+\frac{−3}{\mathrm{s}−2}\\ =\frac{2}{\mathrm{s}+1}−\frac{3}{\mathrm{s}−2}\end{array}$

Therefore, the partial fraction expansion for the given rational function is $\frac{2}{\mathrm{s}+1}−\frac{3}{\mathrm{s}−2}$.